(3x-1)(x-5)=(x+1)(x-1)

Solution for (3x-1)(x-5)=(x+1)(x-1) equation:

Simplifying
(3x + -1)(x + -5) = (x + 1)(x + -1)

Reorder the terms:
(-1 + 3x)(x + -5) = (x + 1)(x + -1)

Reorder the terms:
(-1 + 3x)(-5 + x) = (x + 1)(x + -1)

Multiply (-1 + 3x) * (-5 + x)
(-1(-5 + x) + 3x * (-5 + x)) = (x + 1)(x + -1)
((-5 * -1 + x * -1) + 3x * (-5 + x)) = (x + 1)(x + -1)
((5 + -1x) + 3x * (-5 + x)) = (x + 1)(x + -1)
(5 + -1x + (-5 * 3x + x * 3x)) = (x + 1)(x + -1)
(5 + -1x + (-15x + 3x2)) = (x + 1)(x + -1)

Combine like terms: -1x + -15x = -16x
(5 + -16x + 3x2) = (x + 1)(x + -1)

Reorder the terms:
5 + -16x + 3x2 = (1 + x)(x + -1)

Reorder the terms:
5 + -16x + 3x2 = (1 + x)(-1 + x)

Multiply (1 + x) * (-1 + x)
5 + -16x + 3x2 = (1(-1 + x) + x(-1 + x))
5 + -16x + 3x2 = ((-1 * 1 + x * 1) + x(-1 + x))
5 + -16x + 3x2 = ((-1 + 1x) + x(-1 + x))
5 + -16x + 3x2 = (-1 + 1x + (-1 * x + x * x))
5 + -16x + 3x2 = (-1 + 1x + (-1x + x2))

Combine like terms: 1x + -1x = 0
5 + -16x + 3x2 = (-1 + 0 + x2)
5 + -16x + 3x2 = (-1 + x2)

Solving
5 + -16x + 3x2 = -1 + x2

Solving for variable 'x'.

Reorder the terms:
5 + 1 + -16x + 3x2 + -1x2 = -1 + x2 + 1 + -1x2

Combine like terms: 5 + 1 = 6
6 + -16x + 3x2 + -1x2 = -1 + x2 + 1 + -1x2

Combine like terms: 3x2 + -1x2 = 2x2
6 + -16x + 2x2 = -1 + x2 + 1 + -1x2

Reorder the terms:
6 + -16x + 2x2 = -1 + 1 + x2 + -1x2

Combine like terms: -1 + 1 = 0
6 + -16x + 2x2 = 0 + x2 + -1x2
6 + -16x + 2x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
6 + -16x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + -8x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + -8x + x2)' equal to zero and attempt to solve:

Simplifying
3 + -8x + x2 = 0

Solving
3 + -8x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -8x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -8x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -8x + x2 = 0 + -3
-8x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-8x + x2 = -3

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = -3 + 16

Reorder the terms:
16 + -8x + x2 = -3 + 16

Combine like terms: -3 + 16 = 13
16 + -8x + x2 = 13

Factor a perfect square on the left side:
(x + -4)(x + -4) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(x + -4) equal to 3.605551275 and -3.605551275.

Subproblem 1
x + -4 = 3.605551275

Simplifying
x + -4 = 3.605551275

Reorder the terms:
-4 + x = 3.605551275

Solving
-4 + x = 3.605551275

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '4' to each side of the equation.
-4 + 4 + x = 3.605551275 + 4

Combine like terms: -4 + 4 = 0
0 + x = 3.605551275 + 4
x = 3.605551275 + 4

Combine like terms: 3.605551275 + 4 = 7.605551275
x = 7.605551275

Simplifying
x = 7.605551275

Subproblem 2
x + -4 = -3.605551275

Simplifying
x + -4 = -3.605551275

Reorder the terms:
-4 + x = -3.605551275

Solving
-4 + x = -3.605551275

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '4' to each side of the equation.
-4 + 4 + x = -3.605551275 + 4

Combine like terms: -4 + 4 = 0
0 + x = -3.605551275 + 4
x = -3.605551275 + 4

Combine like terms: -3.605551275 + 4 = 0.394448725
x = 0.394448725

Simplifying
x = 0.394448725

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.605551275, 0.394448725}
Solution
x = {7.605551275, 0.394448725}